A method for computing the autocovariance of renewal processes

In this paper we derive formulae for the autocovariance functions of renewal and renewal reward processes. The derivation is based on a Poissonization technique of a renewal process. The formulae are expressed in the form of Laplace transforms. In some cases we may invert the Laplace transforms analytically, but in general we have to invert them numerically.     

Lifetesting and estimation with arbitrary distribution function

We consider the problem of estimating the life–distribution F from censored lifetimes. The observation scheme is renewal testing over a long time horizon although the results can apply to survival testing with repetitions. We exhibit a product–limit estimator of F which is shown to be consistent and to converge weakly to a GAUSsian process. To do this we first extend these properties of the NELSON-AALEN martingale estimator to the family of PoissoN–type counting processes. Our proof of weak convergence is based on the general functional central limit theorems for semimartingales as developed by .JACOB, SHIRYAYEV and others     

Normal Deviation and Poisson Approximation of a Security Market by the Geometric Markov Renewal Processes

We consider the geometric Markov renewal processes (GMRP) as a model for a security market. Normal deviations of the geometric Markov renewal processes for ergodic averaging and double averaging schemes are derived. We introduce Poisson averaging scheme for the geometric Markov renewal processes. European call option pricing formulas for GMRP are presented.     

Long-Range Dependence of Markov Renewal Processes

This paper examines long‐range dependence (LRD) and asymptotic properties of Markov renewal processes generalizing results of Daley for renewal processes. The Hurst index and discrepancy function, which is the difference between the expected number of arrivals in (0, t] given a point at 0 and the number of arrivals in (0, t] in the time stationary version, are examined in terms of the moment index. The moment index is the supremum of the set of r > 0 such that the rth moment of the first return time to a state is finite, employing the solidarity results of Sgibnev. The results are derived for irreducible, regular Markov renewal processes on countable state spaces. The paper also derives conditions to determine the moment index of the first return times in terms of the Markov renewal kernel distribution functions of the process.     

Inversion of the Renewal Density with Dead-time

Stationary renewal point processes are defined by the probability distribution of the distances between successive points (lifetimes) that are independent and identically distributed random variables. For some applications it is also interesting to define the properties of a renewal process by using the renewal density. There are well-known expressions of this density in terms of the probability density of the lifetimes. It is more difficult to solve the inverse problem consisting in the determination of the density of the lifetimes in terms of the renewal density. Theoretical expressions between their Laplace transforms are available but the inversion of these transforms is often very difficult to obtain in closed form. We show that this is possible for renewal processes presenting a dead-time property characterized by the fact that the renewal density is zero in an interval including the origin. We present the principle of a recursive method allowing the solution of this problem and we apply this method to the case of some processes with input dead-time. Computer simulations on Poisson and Erlang (2) processes show quite good agreement between theoretical calculations and experimental measurements on simulated data.     

Truncated sequential change-point detection based on renewal counting processes II

The standard approach in change-point theory is to base the statistical analysis on a sample of fixed size. Alternatively, one observes some random phenomenon sequentially and takes action as soon as one observes some statistically significant deviation from the “normal” behaviour. The present paper is a continuation of Gut and Steinebach [2002. Truncated sequential change-point detection based on renewal counting processes. Scand. J. Statist. 29, 693–719] the main point being that here we look in more detail into the behaviour of the relevant stopping times, in particular the time it takes from the actual change-point until the change is detected, more precisely, we prove asymptotics for stopping times under alternatives.     

Variance Estimation Based on Invariance Principles

Consistent variance estimators for certain stochastic processes are suggested using the fact that (weak or strong) invariance principles may be available. Convergence rates are also derived, the latter being essentially determined by the approximation rates in the corresponding invariance principles. As an application, a change point test in a simple AMOC renewal model is briefly discussed, where variance estimators possessing good enough convergence rates are required.     

On Vervaat and Vervaat-error-type processes for partial sums and renewals

We study the asymptotic behaviour of stochastic processes that are generated by sums of partial sums of i.i.d. random variables and their renewals. We conclude that these processes cannot converge weakly to any nondegenerate random element of the space D[0,1]$D[0,1]$. On the other hand, we show that their properly normalized integrals as Vervaat-type stochastic processes converge weakly to a squared Wiener process. Moreover, we also deal with the asymptotic behaviour of the deviations of these processes, the so-called Vervaat-error-type processes.     

LIMIT THEOREMS FOR SUBCRITICAL AGE-DEPENDENT BRANCHING PROCESSES WITH TWO TYPES OF IMMIGRATION

ABSTRACT

For the classical subcritical age-dependent branching process the effect of the following two-type immigration pattern is studied. At a sequence of renewal epochs a random number of immigrants enters the population. Each subpopulation stemming from one of these immigrants or one of the ancestors is revived by new immigrants and their offspring whenever it dies out, possibly after an additional delay period. All individuals have the same lifetime distribution and produce offspring according to the same reproduction law. This is the Bellman-Harris process with immigration at zero and immigration of renewal type (BHPIOR). We prove a strong law of large numbers and a central limit theorem for such processes. Similar conclusions are obtained for their discrete-time counterparts (lifetime per individual equals one), called Galton-Watson processes with immigration at zero and immigration of renewal type (GWPIOR). Our approach is based on the theory of regenerative processes, renewal theory and occupation measures and is quite different from those in earlier related work using analytic tools.     

A note on replacement policy comparisons from NBUC lifetime of the unit

Age and block replacement policies are commonly used in order to reduce the number of in-service failures when the systems are functioning indefinitely. In reliability theory, the lifetime of a system can be modeled by means of the NBUC aging class that is characterized throughout comparisons of the residual lives in the sense of the icx order. The purpose of this paper is to establish stochastic comparisons between the age (block) replacement policy and a renewal process with no planned replacements when the lifetime of the unit is NBUC. Supported by Ministerio de Ciencia y Tecnología under grant BFM2000-0362     

Recurrent first hitting times in Wiener diffusion under several observation schemes

Recurrent events are commonly encountered in the natural sciences, engineering, and medicine. The theory of renewal and regenerative processes provides an elegant mathematical foundation for idealized recurrent event processes. In real-world applications, however, the contexts tend to be complicated by a variety of practical intricacies, including observation schemes with different phase and data structures. This paper formulates a recurrent event process as a succession of independent and identically distributed first hitting times for a Wiener sample path as it passes through successive equally-spaced levels. We develop exact mathematical results for statistical inferences based on several observation schemes that include observation initiated at a renewal point, observation of a stationary process over a finite window, and other variants. We also consider inferences drawn from different data structures, including gap times between renewal points (or fragments thereof) and counts of renewal events occurring within an observation window. We explore the precision of estimates using simulated scenarios and develop empirical regression functions for planning the sample size of a recurrent event study. We demonstrate our results using data from a clinical trial for chronic obstructive pulmonary disease in which the recurrent events are successive exacerbations of the condition. The case study demonstrates how covariates can be incorporated into the analysis using threshold regression.     

Bounds of moment generating functions of some life distributions

In this article we show that if a life has new better than used in expectation (NBUE) ageing property and if the mean life is finite then the moment generating function exists and is finite. In fact, the moment generating function is shown to be bounded above by that of the exponential distribution with the same mean. Analogous results are also proven for two much bigger families of life distribution, namely, the new better than renewal used in expectation (NBRUE) and the renewal new is better than used in expectation (RNBUE) and the renewal new better than renewal used in expectation (RNBRUE), provided that the life has finite two moments. Further, stronger results are also obtained for the smaller new better than used version of the above classes.     

Extensions to Basu's theorem,factorizations, and infinite divisibility

We define a notion of approximate sufficiency and approximate ancillarity and show that such statistics are approximately independent pointwise under each value of the parameter. We do so without mentioning the somewhat nonintuitive concept of completeness, thus providing a more transparent version of Basu's theorem. Two total variation inequalities are given, which we call approximate Basu theorems.     

On the Analysis of Quasi-Life Tables

Baxter (1994) defined a quasi-life table in which the data arise from many concurrent, independent, discrete-time renewal processes. The processes are not observed individually, only the total numbers of renewals at each time point are observed. The estimates proposed by Baxter (1994), based on the discrete-time renewal equation, are studied more formally here, and some extensions are made.     

Prediction in trend-renewal processes for repairable systems

Some problems of point and interval prediction in a trend-renewal process (TRP) are considered. TRP’s, whose realizations depend on a renewal distribution as well as on a trend function, comprise the non-homogeneous Poisson and renewal processes and serve as useful reliability models for repairable systems. For these processes, some possible ideas and methods for constructing the predicted next failure time and the prediction interval for the next failure time are presented. A method of constructing the predictors is also presented in the case when the renewal distribution of a TRP is unknown (and consequently, the likelihood function of this process is unknown). Using the prediction methods proposed, simulations are conducted to compare the predicted times and prediction intervals for a TRP with completely unknown renewal distribution with the corresponding results for the TRP with a Weibull renewal distribution and power law type trend function. The prediction methods are also applied to some real data.     

STATIONARITY OF DELAYED SUBORDINATORS

It is shown that, analogous to partial-sum processes in renewal theory, nondecreasing Lévy processes (subordinators) can be delayed such as to show a certain stationarity.     

Diffusion approximations for insurance risk processes

We consider a risk-reserve process for an insurance company where premium income and the claim sum process are modeled as a renewal reward processes. Moreover, dividends are paid out according to a barrier rule. The aim of the article is to establish a diffusion approximation of this model and to compute ruin probabilities (in finite and in infinite time) and other relevant statistics approximately using the limiting diffusion process. We also demonstrate that, under special circumstances, there exists a stationary distribution for the limiting diffusion.     

Estimating the distribution of one-dimensional discrete scan statistics viewed as extremes of 1-dependent stationary sequences

Discrete one-dimensional scan statistics can be viewed as extremes of 1-dependent stationary sequences. A result of Haiman [1999. First passage time for some stationary processes. Stochastic Process. Appl. 80, 231–248] provides approximations of the distributions of extremes of 1-dependent stationary sequences together with sharp bounds for the corresponding errors. We apply this result to scan statistics generated by Bernoulli r.v.'s and to the charge problem.     

Estimation of parameters for trend-renewal processes

Methods of estimating unknown parameters of a trend function for trend-renewal processes are investigated in the case when the renewal distribution function is unknown. If the renewal distribution is unknown, then the likelihood function of the trend-renewal process is unknown and consequently the maximum likelihood method cannot be used. In such a situation we propose three other methods of estimating the trend parameters. The methods proposed can also be used to predict future occurrence times. The performance of the estimators based on these methods is illustrated numerically for some trend-renewal processes for which the statistical inference is analytically intractable.     

Birth-multiple catastrophe processes

Birth-multiple catastrophe processes are analyzed where the birth transition rates are assumed to be constant while catastrophes are distinguished by having possibly different destinations and possibly different transition rates. The transient probability functions of such birth-multiple catastrophe systems are determined. The solution method uses dual processes, randomization, and sample path counting. Solutions are explicit in terms of being a finite linear combination of products of exponential functions of time, t, and nonnegative integer powers of t. The coefficients within this expansion follow a pattern of rational functions of the transition rates.